1. Field of the Invention
The present invention relates to a thin film property measurement method, particularly to a method for determining the film thickness and optical constants of thin films formed on various types of substrates using a spectroscopic ellipsometer, a method for determining the composition of a crystalline compound semiconductor material, and a method for determining the composition of polycrystalline compound semiconductor material.
2. Description of the Related Art
(General Related Art in Spectroscopic Ellipsometry)
A spectroscopic ellipsometer measures the change in polarization between incident and reflected light, and calculates the film thickness (d) and complex refractive index (N=n−ik) from the change in polarization. The change in polarization (ρ) is represented by ρ=tan Ψexp(iΔ), and is dependent upon the parameters such as the wavelength (λ), the incident angle (φ), the film thickness, the complex refractive index, and accordingly, the relationship between these parameters can be represented by the following expression.(d, n, k)=F(ψ, Δ, λ, φ)
In case of single wavelength ellipsometer, if the incident angle is fixed, only two independent variables of three unknown values of (d, n, k) can be measured, and accordingly, there is the need to fix one of d, n, and k as a known value. Note that in the event that measurement is made with multiple incident angles, the number of measured variables increases, even if the single wavelength ellipsometer is used. However, measured pairs of (Ψφi, Δφi) corresponding to different incidence angles (φ), are partly correlated, leading to difficulties in obtaining precise values of d, n, and k.
The measured spectrum measured by spectroscopic ellipsometer (ΨE(λi), ΔE(λi)), which represents the change in polarization due to reflection from single-layer or multi-layer thin films formed on a substrate, includes all information with regard to n and k of the aforementioned substrate, and d, n, and k of each layer. However, the single combination of the information with regard to n and k of the aforementioned substrate, and d, n, and k of each layer, cannot be simply extracted from the aforementioned measured spectra (excluding the case of semi-infinite substrate). In general, the method for extracting of the aforementioned single combination is referred to as “spectroscopic ellipsometry data analysis”. During this analysis, modeling is performed using the information with regard to n and k of the aforementioned substrate, and d, n, and k of each layer. The information regarding to n and k of the substrate and each layer included in the model is obtained from reference data (known table data), a dispersion formula, or optical constants of a single-layer thin film from a similar material.
The dispersion formula represents the wavelength-dependency of the dielectric constant of the material, wherein the dielectric constant ε(λ) can be determined in the optical range between near infrared light and ultraviolet light based upon the atomic structure of the material. Known examples of dispersion formulas include a formula based on classical physics (a harmonic oscillator), a formula based on quantum mechanics, an empirical formula, and the like, which generally include two or more parameters. The model is applied to the measured data by adjusting all the unknown values (thickness of each layer, parameters of the dispersion formula, volume fractions of material's components, or the like) included in the aforementioned model. This processing is referred to as “fitting”, wherein the thickness, parameters of dispersion formula, the volume fractions, and the like, of each layer are obtained. The dielectric constant ε (λ) of the material can be calculated from the parameters of the dispersion formula, based upon the fitting results. The relation between the dielectric constant of the material and the refractive index is represented by the following expression.ε=N2
Now, brief description will be made regarding fitting operation frequently employed in methods according to the present invention.
(Description Regarding the Fitting Figure of Merit χ2)
With the set of N pairs of measured (experimental) data as Exp(i=1, 2, and so on through N), and with the set of N pairs of the data calculated using the model as Mod(i=1, 2, and so on through N), making assumption that error of measurement follows normal distribution, and with the standard deviation as σi, the mean square error (χ2) is represented by the expression
      χ    2    =            [              1        /                  (                                    2              ⁢              N                        -            P                    )                    ]        ⁢                  ∑                  i          =          1                N            ⁢                                    (                          Exp              -              Mod                        )                    2                /                  σ          i          2                    wherein P represents the number of the parameters. The aforementioned expression indicates that the smaller χ2 is, the better the model matches the measured results. Accordingly, the best model can be selected from multiple models by selecting the model having the smallest χ2.
The aforementioned change in light polarization is proportional to the volume through which the light passes (The phase angle (β) multiplied by the area of the beam's cross section). The phase angle (β) is represented by the following expressionβ=2π(d/λ)(N2−NA2 sin 2φ0)1/2wherein NA and N represent the complex refractive indices of the ambient and substrate respectively.
Making an assumption that the beam's cross section is constant, the change in polarization can be expressedChange in polarization∝Film thickness (d)×Complex refractive index (n)×f(φ)wherein φ represents the incident angle.
As can be understood from this expression, in the event that the film thickness (d) and the complex refractive index (N) are small values, the change of the phase angle (β) becomes small, and the measurement might become difficult.
Furthermore, as can be understood from the aforementioned expression, the precision of the incident angle affects the change in polarization. Accordingly, a method for obtaining a precise incident angle is necessary. That is to say, determination of the precise incident angle allows the precise determination of the change in the polarization of reflected light.
Furthermore, there is great demand for a method for obtaining the precise composition ratio of the compound semiconductor layer, or a method for maintaining the composition ratio in a predetermined range. The complex refractive index N of a compound semiconductor layer A(1-x)Bx is determined, depending on the value of x. For example, in a case of a compound semiconductor layer A(1-x)Bx formed by atoms A and B on the substrate A, and furthermore in the event that the composition ratio x is small, the difference between the (n0, k0) of the substrate and the (nj, kj) of the compound semiconductor layer is almost non-existent, leading to small change in polarization, due to this layer. Accordingly, the inventors of the present invention believe that precise measurement of the incident angle is important (see FIGS. 18 and 19).
Recently, there is great demand for measurement of the concentration or the like of the atoms (atoms of interest) in a desired polycrystalline compound semiconductor layer.
In the present invention, Effective Medium Theory (EMT) is used to calculate the effective dielectric function of materials, those dielectric function's wavelength dependence is difficult or impossible to express, using only one dispersion formula.
In general, the effective dielectric constant (ε) of the host material which contains N number of inclusions (guest materials), each inclusion layer enough to possess it's own dielectric constant, is represented by the expression
            (              ɛ        -                  ɛ          h                    )        /          (              ɛ        +                  k          ⁢                                          ⁢                      ɛ            h                              )        =            ∑              j        =        1            N        ⁢                            f          j                ⁡                  (                                    ɛ              j                        -                          ɛ              h                                )                    /              (                              ɛ            j                    +                      k            ⁢                                                  ⁢                          ɛ              h                                      )            wherein εh represents the dielectric constant of the host material, εj represents the dielectric constant of the j-th guest material, and k represents a screening factor.
Now, let us consider a case in which one cannot distinguish between the host material and the guest material, i.e., a case that materials of comparable amount have been mixed. In this case, approximation can be made wherein the dielectric constant of the host material and the effective dielectric constant of mixed material are the same εh=ε, therefore εh in the aforementioned expression is replaced by the effective dielectric constant ε. The aforementioned approximation is called “Bruggeman Effective Medium Approximation”, which will be simply referred to as “EMA” in this specification hereafter. Using the EMA, the effective dielectric constant ε of a material, wherein three spherical components a, b, and c have been uniformly mixed, is obtained by the expressionfa(εa−ε)/(εa+2ε)+fb(εb−ε)/(εb+2ε)+fc(εc−ε)/(εc+2ε)=0wherein ε represents the effective dielectric constant which is to be obtained, εa, εb, and εc, represent the dielectric constants of the spherical components a, b, and c, respectively, and fa, fb, and fc, represent the volume fraction of the corresponding components. Volume fraction will be referred to as “Vf”, hereafter. Note that fa+fb+fc=1.
Effective Medium Approximation (EMA) is applicable, if the separate regions (components) of mixed material are small compared to the wavelength of light. EMA is used to model thin film on substrate, if this film is either microscopically inhomogeneous or discontinuous or formed by several physically mixed materials.
Now, description will be made regarding a case that the materials a, b, and c have been mixed. In this case, EMA is used to calculate the dielectric constant of the mixed layer from the volume fractions of each component and the dielectric constants of corresponding materials a, b and c. Dielectric constant of each component can be determined by either reference data or dispersion formula. Assuming the mixed layer thickness, model can be built and fitted to the measured data.
Crystalline material as used here means a single crystal, which can be regarded as being formed of a single infinite grain (having no grain boundary). On the other hand, actual polycrystalline material is formed of a great number of single crystals (grains), or a great number of grains and amorphous components (material may contain cavities depending on the manufacturing method). To facilitate description, in general, polycrystalline material can be regarded as a mixture of the crystalline and the amorphous components, or a mixture of the crystalline, amorphous, and void components. Accordingly, the dielectric constant of the polycrystalline material can be calculated using the aforementioned EMA.
Dielectric constants are well known for various crystalline materials, and in general, known data (reference) is used. On the other hand, the dielectric constant of the amorphous material is greatly influenced by the manufacturing method, and accordingly, reference data exists and can be used only for limited materials manufactured by limited number of methods.
With the polycrystalline material, the size of grains, the size of grain boundaries, the presence or absence of the amorphous component, and the crystallization ratio have a great influence upon the dielectric constant thereof. Accordingly, reference exists and can be used only for limited materials manufactured by limited number of methods.
For materials which are commonly employed in the semiconductor industry, such as silicon, reference exists and can be used not only for crystalline silicon (c-Si), but also for amorphous silicon (a-Si) and polysilicon (p-Si).
Dielectric constant reference exists and can be also used for crystalline SiGe (c-Si1-xGex), which has recently come into great demand, with various Ge concentrations (x). The dielectric constant of the crystalline SiGe (c-Si1-xGex) is dependent on the Ge concentration, and accordingly, the Ge concentration can be calculated by obtaining the dielectric constant of the c-SiGe from the spectroscopic ellipsometry data.
On the other hand, polycrystalline SiGe (p-Si1-xGex) can have not only various Ge concentrations but also various crystallization ratios, as described above. Accordingly, the dielectric constant of the polycrystalline SiGe (p-Si1-xGex) is influenced by the Ge concentration and the crystallization ratio thereof. Thus, it is difficult to make a reference for the polycrystalline SiGe (p-Si1-xGex).
As described above, the Ge concentration of p-SiGe cannot be calculated based upon the dielectric constant calculated from spectroscopic ellipsometry data using a reference in the same way as with c-SiGe described above.